Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.

Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential $${v(x)= \epsilon \chi(x) |x|^{-1}}$$, where $${\epsilon}$$ is sufficiently small and $${\chi \in C_0^{\infty}}$$ even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper.